Failure of completeness properties of intuitionistic predicate logic for constructive models

D ANIEL L EIVANT

Failure of completeness properties of intuitionistic predicate logic for constructive models

Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 60, série Mathéma- tiques, n

^o

13 (1976), p. 93-107

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© Université de Clermont-Ferrand 2, 1976, tous droits réservés.

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FAILURE OF COMPLETENESS PROPERTIES OF INTUITIONISTIC PREDICATE LOGIC

FOR CONSTRUCTIVE MODELS Daniel LEIVANT

University of Amsterdam, AMSTERDAM, Netherlands

ABSTRACT

We consider a

principle

^of

constructivity

^{RED which states} that every decidable

predicate

over the natural numbers is

(weakly) recursively

enumerable

(r.e.).

^{RED is}

easily

^{seen to be}

derived from Church’s thesis

CT 0 ^(every

construction is

given by

^{a recursive}

function»).

Results :

(1)

^RED

implies

^{that the}

species

^of valid first order

predicate

^{schemata is not r.e., and}

hence - that intuitionistic first order

predicate logic .t 1 is incomplete.

(2)

^{We construct a}

specific

^{schema of}

£1

^{which is valid if RED , but}

unprovable (3)

^{The two results above hold even when}

validity

^is

generalized

^to

validity

^with

Kreisel-Troelstra

[ 70 ]’s

^choice sequences ^as

parameters.

(4)

The method is used also to construct a schema

of 9, 1 , unprovable

^{in L}

1, but

^of

whose all metasubstitutions

with L 0

number theoretic

predicates

^are

provable

ⁱⁿ

1 p

Heyting’s

arithmetic G. This is a

simple

^{bound on}

possible improvements

^{of the}

maximality

result of Leivant

[ 75 ] .

1. INTRODUCTION

For a

general

introduction to the

subject

^{matter of this note} the reader is referred to Troelstra

[76 ] .

’

The main two results of this note are the

following. Firstly,

^{we show}

(along

^{the line of}

Kreisel

[70] ] p. 133)

^{that the schema RED} mentioned above in the abstract

implies

^{that the}

species

^of

intuitionistically

valid formulae of

predicate logic

^{is not r.e.. This we do}

by exhibiting

^a sequence

( F )

^of

^formulae,

^the

^validity

^of

Fm~n meaning intuitively :

othe r.e.

species Wm ^and Wn

^{are not}

separable by

any

pair

of decidable

species». By

^RED

decidable

species

^are

(weakly)

^{r.e., so we} may say ^instead

Wm

^and

Wn

^are

recursively inseparable».

^The

proof

^is then concluded

by showing (intuitionistically)

^{that the}

species

((m,n) ~ I _{Wm and Wn}

^are

recursively inseparable)

^{is not r.e..}

Secondly,

^we

pick

up ^m,n for which

Fm n

^{is not}

^valid,

^{but s.t.} the recursive

inseparability

^of

Wm

^and

Wn

^is

^provable

in intuitionistic arithmetic. This last

proof

may be

reshaped

^to

apply

^to any

given 10

^{model of}

arithmetic ;

^{we conclude that} _any

1

10

-metasubstitution of

F m n

^is

^provable

ⁱⁿ

arithmetic,

^thus

proving

^result

⁽⁴⁾

^{of the}

I ^’

abstract.

1.1. NOTATIONAL CONVENTIONS.

We use the

logical

^constants

&#x26;,

A , -~ ,

rr , rq

^{and .1}

(for absurdity ; negation

^-,

is then

definable).

^{The letters}

P,Q,T,Z,S...

^{are used for}

predicate parameters (in

^the

language

of first-order

predicate logic 1),

while bold face T,Z,S... ^are used for certain number theoretic

predicates explicitely

defined below. ^’

For recursion theoretic notions we use the notations of Kleene

[ 52 ]

^and

[69 ]

^: Tn for the

computation predicate

for n-ary

functions,

^{U for the}

result-extracting

^function,

~ e~

^{for the e’th}

_partial

recursive

function, "

^for

partial equality,

^etc.

We assume

given

^an encodement of sequences

i.e. -

(n~,

^...,

nk) ^{2013~ (}

no, ^...,

nk) ^together

^{with a}

length-function

lth and inverse

projection

^functions

(n) Iwhich ^satisfy

We also fix some

G6del-coding

^for

syntactic objects,

^and write ’ a ’ for the code of a .

For formal ^r.e. theories

ct

we write

Prov ^(p,r)

^{for a canonical}

proof predicate

^for

Cc ,

and

Pr (r)

^{for 7} p Prov

(p,r).

1.2. -1.4. FORMAL SETTING.

1.2. LANGUAGE.

°

Our metamathematical

setting

^is the intuitionistic

theory

^of

species, ~ 2.

^An

^inspection

on the

proofs

below shows that

actually only

^{a small}

fragment

^is

^used,

but since it is of little value for our purpose ^{to delimit} this

fragment precisely

^{we do not bother to do so.}

Prawitz

( [ 65 ]

_p.

72)

has shown that

Heyting’s Arithmetic Ci-

is

interpretable

however,

^to avoid notational

complications

^{we shall use lower case}

^a,b,c

^etc. for numeric

variables,

^and lower ^case x,y,xi ^etc. for unrestricted first order variables. A list zo, ^..., zi, ... ^of unrestricted

variables will be reserved for a

specific

^{use, and we shall assume that}

they

^{do not occur in}

formulae otherwise. We also assume that G as well

as L2

^contain

^symbols

^and

^defining equations

^{for all}

primitive

^{recursive functions.}

The

system

^we

study

^is first order intuitionistic

predicate logic .9

l’ without

equality

and without function

symbols.

^{We shall find it} convenient to refer also to extended to the

language

^{with all constants of G}

( = , numerals

^and

prim.

^rec.

functions).

^{We write}

1 G

^for

the

resulting system.

1.4. THE VALIDITY PREDICATE.

Tarski’s definition of a

II 1

₁

validity predicate

^{Val for £}

1(analogously

1 ^{to the} classical

case)

^{is standard}

(cf.

e.g. ^Tarski

[36 J ).

^{The central}

property

^{of Val is of course}

for each schema G where P is the list of all

predicate parameters occurring

ⁱⁿ

^G, D 1

is a

(unary) predicate

variable and

GD

comes from G

by restricting

^all

quantifiers

^{to D.}

To avoid confusion one should note that the

predicate

^{Val does not} express any

analysis

^{of the notion of} constructive truth

(contrary

^to truth definitions like Beth’s and

Kripke’s

^{semantics and the various}

realizability predicates).

^{Here the} constructive content of

! 1

^shows

only through

^the constructive

meaning given (in S- ₂₎

^{to the}

logical

^constants

occurring

^{in Val.} Our discussion is therefore valid

regardless

of any

specific analysis

^of

constructivity.

1.4, THE SCHEMA RED.

We define the schema RED, for orecursive

enumerability

of decidable

predicates»

^{to be}

It is

easily

^{seen that RED is derivable from the}

following

^weak variant of Church’s thesis

CTo :

V. Lifshits has ^sown

(in ^1974, unpublished)

^{that even the}

positive

^version

CT !

^{of I}

is

strictly

^weaker

(in G )

^than

CT 0 .

2. CHARACTERIZATION OF (WEAKLY) RECURSIVELY INSEPARABLE PAIRS OF R.E. SETS VIA RED.

2.1. A FINITE AXIOMATIZATION OF KLEENE’S PREDICATE T.

In the

language ^{of ~} l, fix

^three

predicates Eq(x,y), ^Z(x)

^and

S(x,y).

^{We think of}

these as

representing equality,

^{zero and the successor relation. Let}

AS _{( *}

the axiom of the

theory

^of

successor)

^{be the}

conjunction

^of the closure of the

following

^formulae

Consider the familiar

definig

^{schemata for}

primitive

^recursive functions :

These

equations

may be axiomatized

by

The characteristic function ^T of Kleene’s

predicate

^{T is}

primitive recursive ;

there is therefore a list

e

^~~~,

e.

of

equations,

^{each of which is an} instance of one of

(1) - (5)

^{with some}

auxiliary

^function

letters

f ,

^..., t , which defines ^{T .}

Taking

^the

corresponding

instances of

(1) * - (5) *,

with

predicate

^letters

^{F ,}

₀ ^...,

Fk,

^T

corresponding

^to

^{f ,}

₀ ^{..., t , we obtain a list}

B ,

^...,

^Bm

of formulae which defines T. We let

where

Ci

^{is the} closure of

Bi ⁽ⁱ

^L

^m).

^{Let z ,}

..., zi, ...

^{be a list} of variables

(not occurring

0 in

A) ;

^define

2.2. We refer below to the schema

Write

(in

^the

language

^{of £}

2)

where W e

is the number theoretic

predicate

o

Val w, E1 (rG(a1,a2)7)

then reads : the r.e. sets

W a 1

^and

Wa2

^{are not}

^{separable by}

^any

^pair

of decidable r.e. sets.

2.3. LEMMA.

(in ~~)

where n : =

max [ a,b,c ].

PROOF : One proves

for each

primitive

^recursive

predicate Fi

^{used in} the definition of T,

by

^{induction on the}

length

^of definition of

Fi.

The details are routine. 0

2.4. LEMMA.

(in £ 2)

Hence

where Z, ..., ^{T are} the number theoretic

predicates

^{for zero, successor etc.}

Since the

premise

^of

(1)

^is

trivially

^{true, we obtain}

^Val~

^as

required.

II. Assume

Fix P and

Q

^{and assume}

and

From

(2)

^and

(4)

^{we then}

get

Assume

which, by

lemma 2.3. and

assumption (3) implies ^{E [ T,P,Q ]} (zm,zn).

^This

^being

^{derived from}

(6),

^we

get by

intuitionistic

propositional logic

^that

(5) implies

^{-~2013 E}

[ T,P,Q ^] (z , z~)

as

required. o

2.5. LEMMA.

(in £ 2&#x3E;

^RED

^implies

PROOF : The

implication

^{from left to}

right

is trivial. Assume on the other hand

and, fixing

^{P and}

^Q,

^{assume D}

^{[P,Q ].} By

^{RED D}

[ P,Q] implies

^the weak existence of ^some el,

e2

^s.t.

which

by ⁽¹⁾ implies ,IE [1 ,P,Q] I (m,n)

^as

required. 0

2,6. PROPOSITION,

(in - 2)

^RED

^implies

PROOF : Immediate from 2.4 and 2.5. ~

3. WEAK INCOMPLETENESS

OF I ,

^{(UNDER RED).}

,

/

w,E01

3. 1. PROPOSITION. The

species

^{S :}

= ^{m,n) ( Val 1}

^{is not r.e.}

PROOF : Assume that the

speciesS

^{is r.e., i.e. - for some}

primitive

^recursive

predicate

^lI.

Let

where

Note that we may define ^a

primitive

^recursive

proof predicate Prov C+

^for

^{G as}

^{follows :}

where

imp

^{is a}

primitive

recursive function which satisfies

and where

Prov ~

^is

easily proved

^{in G to}

satisfy

^the

elementary derivability

conditions. Let G

where neg ^{is a}

primitive

^recursive function which satisfies

neg( r F ’ ) = ~ 2013!F .

_{for any} formula F in the

language

^Since

G

_{is r.e., the}

consistency ^{of G} is equivalent

^to

the statement

oWa 1 and ^Wa 2 are recursively inseparable» ^(see Smullyan ^{[ 61 ]}

p. 6~. thm.

27 ;

notice that the

proof

^{there is}

intuitionistic).

Put

formally :

where Con + : = -

Pr +(r_l_7).

^In

£2

^we know however that Con ₊ is true,

20132013C G ² G

and

consequently Va l’ a 2

^{must be an axiom}

^{of Ct,+. Together}

^with

⁽²⁾

^this

^implies

Con - +

which contradicts G6del second

incompleteness

^theorem

(since Prov ,

satisfies the

elementary

derivability conditions).

^{Thus the}

assumption

^{that S is r.e. leads}

2)

^{to a} contradiction. 8 REMARK : The reference to

Smullyan’s

^{theorem is made for the sake of}

brevity.

One may instead

pick

up any

pair We , We

_{1 2}^{of r.e. sets which are}

^proved

^{in G to be}

recursively inseparable (cf.

e.g.

Rogers ^{[ 67 1}

p. 94 thm.

XII(c),

^whose

proof

^is

intuitionistic).

Defining

we find that

Con

^is

equivalent (in C)

^{to the recursive}

inseparability

^of

Wd

^and

Wd2.

C

^{1 2}

THEOREM I.

(Weak incompleteness).

^In

L2

^{+ RED we can} prove that ^not every valid formula is

provable.

PROOF : If every valid formula of

£, 1

^is

^provable

^{in ~ i then the}

^species

^{of valid} formulae is the same as the

species

^of

provable ^formulae,

^{and so it must be r.e.. The} formulae of the form

An - G(zm, zn)

^are

recursively recognizable,

^{and thus the} valid formulae of this form make _up

an r.e.

species

^{as well.}

By

2.6. S is then _r.e.,

contradicting proposition

:t 1. above.

3.2. The result of 3.1 can be

classically improved by

^the

following

PROPOSITION :

Wm, Wn rec. inseparable)

^is

II2-complete.

o

PROOF.

(essentially

^{due to C.}

Jockusch)

^{Fix a}

pair Rl, R’2

^of

recursively inseparable

^{r.e. sets.}

Let

k(e), hl(e), h2(e)

^be

^prim.

^rec. functions defined

(through

^{the s-m-n}

theorem) by

Then :

(i) We

^finite

(ii) We

^infinite

So the ^set

reduces to S. But I is known to be

II02-complete (cf. Rogers ^[67] p. ^326, 1.3),

^{and hence}

0

S is also

n 2-complete. 8

3.3. PROPOSITION. There are numbers _m,n for which we can prove ^{+ RED} that

An "4 G(zm, zn) is

^{valid but not}

provable in ¿ 1.

P ROOF :

By

^the

proof

^{of 3.1. there are m, n s.t. and so}

by

^2.6

An ~ G(zm, zn)

is

valid, assuming

^{RED. On the other}

^hand,

^if

then

(1) ~ ^G[

^T,P,Q

^{] (m,n)}

^{for any unary} number theoretic

predicates

P,Q.

The G6del translation

Go

of G

(see

^Kleene

[ 52 ]

p.

493)

^{is therefore also}

provable

ⁱⁿ

^{C ;}

but Do is

provable (idem, p.119 *51 _a),

and

by

intuitionistic

predicate ^{logic E[ T,P,Q]}

⁰

implies

^{--7 --iE}

^[

^T,P,Q

^whenever

^{P and Q do not}

contain Y ,

^so

(1) implies

which

implies (in ~ 2)

^{that -T E}

^[ T,WM,Wn] ^(m,n)

^{is true.}

^For

^{the m,n} considered

Wm

and

wn

^{are however}

(provably) disjoint,

^{and so}

by

the definition of E we have -,E

[T,Wm lwn ^(m,n),

^a contradiction. D

REMARK : A

simpler proof

^{of the above}

proposition

^{runs as follows.}

Classically

the schema G is

equivalent

^{to E. E is}

obviously

^{not valid}

classically,

^{so it cannot be}

provable

^{even in classical}

first order

logic, by

^G6del’s

completeness

^{theorem. The} formalization of this

proof

^{in the}

intuitionistic

theory

^of

species

^{is however}

slightly problematic.

3.4. INCOMPLETENESS W.R.T. VALIDITY WITH CHOICE PARAMETERS

Let a be a variable for choice sequences, ^{and assume} that for the kind of choice sequences considered ^we have

which is the case for the choice sequences

investigated by

Kreisel-Troelstra

r

⁷⁰

] (cf.

^6.2.1

there).

(1) implies quite trivially

^{for every}

negated

^formula

-,A(a )

Let us refer now to a notion of

validity

with choice

parameters,

where

Ga

^comes from G

by replacing

^{each atomic} subformula

P(tl,

^...,

tn) by

P(a , tl,

^...,

tn)- ^(This

^{is not weaker than}

allowing

the choice

parameters

^{to be distinct for} each

predicate letter,

^{as can}

readily

^be

seen.)

A

straightforward

observation shows now that the discussion above

remains

^correct

cs w,CS w

when Val and

Valw

^are

replaced by

^{Val and an}

analogue Val w,GS respectively,

^{Val 1}

remains

unchanged

^and the schema RED is

generalized

^to

But

by (2) REDCS

is

implied outright by

^{RED, since RED is}

negative,

^and

(~-3)

^{with a}

varying

^over total recursive functions is

just

^a

special

^{case of}

^(the quantified

^variant

^of)

^RED.

To

recapitulate,

^we have obtained

THEOREM II. In

~2

^{+ RED we can} prove that the

species

of formulae

(of ~:1)

^{which are}

valid with choice

parameters

^{is not r.e., and we} may exhibit ^a

specific

formula of 14 ₁ which is valid with choice

parameters

^{but not}

provable in

4. ~ 1

^{IS NOT}

4.1. DEFINITION OF MAXIMALITY.

Let ^C be a class of number theoretic

predicates

^{and. be a formal}

system

^{in a}

language extending

^the

language

of arithmetic. A formula

H[ Po,

^...,

Pk ^J

^of

11

^{is a}

C-schemaof 3 when

for every

Po-

^{I ..., ...,}

* pk in C. We define formally for 8 = a,C = the class of E

^{in C . We define}

formally

^{for ’)}

= G ,

^{C = the class}

of 2

_I

predicates :

where (for

^m~

1)

(In defining

^the

predicate

^{Sch we}

implicitly

^{use a}

primitive

^recursive

substitution function, the

definition of which is

routine).

A

system L

^of

logic

^{is said to be C -maximal}

^{for a}

^if

~ = ~ H ^{( H}

^{in the}

language ^{His -}

^{C -schema of}

S . }

de

Jongh

^has

proved (in 1969)

^{that the} intuitionistic

propositional logic

^is

E01-maximal

for intuitionistic

arithmetic G (and

certain extensions of

C). 141

^Leivant

1

1

(chs. B3-B5)

^{it is}

proved

^that

£1

^{is IT}

2-maximal

^{for G}

^(and

^certain extensions

of C ).

We shall now show that - is ₁

not E01-maximal

_1, ^even for a

fragment ^{of G .}

^.

4.2, For a schema H =. H

[Z,S,Eq,F 0’

^...,

^{Frg T ;} P

^...,

^{Pt ] Of 1 we define}

^the

^schema

G ,

^{and the schema}

of L1G,

^{where Z, ..., T are} the intended number theoretic

predicates,

^{i.e. -}

Z(x) :

^{= x =}

0,

^etc.

If the ^r.e.

interpretations

of the zero, successor and

equality predicates satisfy

^{the axiom of successor}

AS (cf. 2.1),

^then

Z ~

and

S ~ _generate

^{a structure}

isomorphic

^to

through

^{a recursive}

(total) function,

defined as follows :

We let then

For ^a formula H of G we now write

for

the sdwma

of jiG

I 1I1I1t’. II I

,

by restricting

^all

quantifiers

^to

N d’

4.3. I,EMMA.

(in G )

^Let

where n : = defined as in 2.], but with defining formulae for all 1

predicates TJ , where j

ranges ^over the dimensions)) of the

pre()1(’" If’

^{iii I I .}

PROOF : Let G be given by a Gentzen natural deduction system (rl.e.g

PraBBit7 ! 7! !).

ForaformulaH ⁼

I (n

₀

...,nm) of

^{G for}

H [Z,S,Eq,..., T] (zno, ...znm]

^{as defined in} 4.2. above. I.e. -

It is easy ^to

verify

^{that for} each inference rule of the (;entzen ~) ^{...1, 111}

(where

^{n , ...,}

np

is the list of all numerals

occurring

ⁱⁿ the formulae shown) the

0 P

isaderivedruleofG+

An d, where

^{n:= max P}

if a formula 11(ii0 , ,...,n )of

^G

is derivable by a natural deduction A

then the tormnh

0 P

H[d I Id (zn 0

occurring

^{in A .}

By

the existential

conjuncts

of the axiom Xs t, (.). ^and (B) ^{in 2 i} q may be ^cut down to n : ^:=~ max

I

^{i ~}

_pl.

^.

Assume now the

premise

^of

(1),

and fix d and e.

For j -

^{t let the}

predicate

^letter

p.

^{be In.. I}

^pLH’1

^{i iiil d,’l’íllt} iliiilil.t ^I

J J

e*= ((e*) 0 ,

₀ ^...,

(e*)t &#x3E; ^through

^{the s-m-n theorem}

by

where the

function v ,

^which

depends

^on

^d,

^{is defined as in 4.2. above.}

Further,

^let _{e 1}

= « e 1) o

₀

,..., (e 1 )t)

be defined

by

By (2)

^now

and so

by

^{the above}

argument

where n : ⁼ max

[nj|j

_J p

J .

^{It is}

readily

^{seen however that for}

where

for j ~

^t

Here

T d

^{is the}

d-interpretation

^{of the}

predicate Ti

^{i.e. -}

_{T d :}

^z

W(d)a

^{for some q.}

by

the definition of the

interpretation generated by ^d, provided A n d

^holds

Since in H

d ~ all quantifiers

^are indeed restricted to

N ,

^we thus have from

(6)

(provided And ^),

^{and so}

^{(5) implies}

^the

provability

^{in Ci of}

This

being

^{the case for}

arbitrary

^{d and e we} have obtained the concliisinn of

(1 ),

^as

4.4. THEOREM III.

£. 1

^is

not E01-maximal for G . Specifically -

^{are m and n f’or whiuh}

AT, 1.4 ^{E [ T,P,Q ] zn)}

^is

a 2: 0 -schema

^of

^{C ,}

^{but is not}

^{provable iii Y, 1. (Here}

^the

schema E is defined as in

2.2.).

PROOF : Let

W~, Wn

^{be a}

^pair

^{of r.e. sets,}

proved

^{in G to be}

recursively inseparable ^(as

ⁱⁿ

I.e. - for each

e0, ^el

which

by

^{lemma 4.3}

implies

^{that for} any

d,e

But E is

existential,

and therefore

I implies (in predicate

^{tB. So}

⁽¹⁾

^for

arbitrary

o

d,e actually

^{states that}

An -&#x3E; E(zm, zn) 01-schema

^{of L .}

On the other hand this formula cannot he

proved ^~n

^{as vBc seen in}

REFERENCES

KLEENE,

^S.C.

[52],

Introduction to

Metamathematics, Wolters-Noordhoff, Groningen, 1952.

[69], Formalized

^Recursive Functionals and Formalized

Realizability,

Memoirs of the AMS 89

(1969).

KREISEL,

^G.

[70],

Church’s thesis : a kind of

reducibility

^{axiom of} constructive

mathematics ;

in lntuitionism and

Proof Theory. (eds. ^Kino, Myhill, Vesley) (North ^Holland, Amsterdam, 1970),

pp. 121-150.

KREISEL,

^{G. and} TROELSTRA, A.S.

[70],

^Formal

systems

^{for some branches} of intuitionistic

analysis ;

^Annals

of

Mathematical

Logic

¹

(1970),

pp. 229-387.

LEIVANT,

^D.

[75], Absoluteness of

Intuitionistic

Logic ;

^PhD

dissertation, University

^of

Amsterdam

(Mathematisch Centrum, Amsterdam, 1975).

PRAWITZ,

^D.

[71 ],

Ideas and results of Proof

Theory ;

ⁱⁿ

Proceedings of

^{the Second} Scandinavian

Logic Symposium (ed. Fenstad) (North ^Holland,

^Amsterdam

1971),

pp. 235-307.

ROGERS,

^H.

[67],

The

Theory of

Recursive Functions and

Effective Computability (McGraw-Hill,

^New

York, 1967).

TARSKI,

^A.

[36] ,

^Der

Wahrheitsbegriff

^{in den}

formalisierten Sprachen,

^{Studia Phil.1}

(1936)

261-405.

English

translation in :

Logic, Semantics, Metamathematics,

Clarendon

Press, Oxford, 1956, pp. 152-278.

TROELSTRA,

^A.S.

[76] , Completeness

^and

validity

for intuitionistic

predicate logic ;